Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space.

In this case, the answer to both questions is yes.
It follows from a more general result:

If $G$ is a sheaf and $f\colon P\to Q$ is a local isomorphism of presheaves
(i.e., a morphism that induces an isomorphism on all stalks),
then the induced map
$$\def\Hom{\mathop{\rm Hom}} \Hom(Q,G)→\Hom(P,G)$$
is an isomorphism.

This abstract result applies to the two cases under consideration
because the maps $F^+→F^\sharp$ and $F\to (F^\sharp)^+=F^\sharp$
are local isomorphisms.

Indeed, even more generally, the natural map $F\to F^+$ is a local isomorphism
for any presheaf $F$.
This follows immediately from the fact that the stalk functor
is cocontinuous, in particular, it preserves the colimit used to define $F^+$.